Atmospheric temperature prediction is fundamentally a physical problem governed by deterministic equations. This paper introduces a hybrid architecture, Pi-ConvLSTM, to investigate how physical principles can guide neural network design and refinement for forecasting regional temperature variations by embedding existing physical knowledge into optimization.In the experimental framework, Pi-ConvLSTM achieves higher Structural Similarity Index Measure (SSIM) than the baseline ConvLSTM across the four forecast time steps, indicating improved performance and temporal stability. Significant reductions are also observed in the mean squared error (MSE) and mean absolute error (MAE). Moreover, analysis of the final loss across different layer counts identified the optimal number of setting layers as a hyperparameter, consistent with theoretical results indicating that the appropriate number of setting layers in this study is no more than two.Notably, Pi-ConvLSTM performs better over ocean than over land, a tendency predicted by analysis of the governing physical equations. Comparisons of SSIM and MSE indicate that state-of-the-art models such as GraphCast, FourCastNet and PanguLM still exhibit this limitation.
The semi-discrete schemes for hyperbolic conservation laws on Cartesian grids are revisited. For the spatial discretisation the traditional 2nd-order slope limited MUSCL schemes are used. We focus on the explicit Runge-Kutta (RK) type methods of order two and three for the temporal discretisation, including those possessing the strong stability preserving (SSP) property. The qualitative and quantitative influence on the computed solutions of the effective spatial stencil size stemming from particular time integrators is analysed. The reduction of the spatial stencils size by freezing the MUSCL slopes for the entire time step is shown to significantly improve the stability and accuracy of the 2nd-order RK methods: SSPRK22 and midpoint rule (MIP) method. The MIP method with frozen slopes (FS), which can be interpreted as a semi-discrete, unstaggered variant of the MUSCL-Hancock scheme, can be used beyond the hyperbolic problems and is shown to be attractive alternative to SSPRK methods. Importantly, this holds for a wide range of the Courant numbers, including values close to the stability limit, while at low Courant numbers the performance is very similar to SSPRK-based schemes. On the other hand, the MIP method does not exhibit the SSP property; therefore we propose a new one-parameter family of continuously differentiable 2nd-order total variation diminishing slope limiters expressed by a polynomial, which allows to control the required level of numerical diffusion to reduce (or eliminate) the spurious oscillations. Within the use of the new slope limiter no stability issues were found when MIPFS method was applied to a demanding fluid flow simulation involving strong shocks. When a very low Mach number flow is considered, the MIPFS method requires less diffusive slope limiter than SSPRK33 to provide convergence. Importantly, this improvement is obtained when simple and universal spatial discretisation is applied: incomplete approximate Riemann solver of HLL-type along with low-Mach corrected MUSCL reconstruction of velocity.
This paper develops unconditional stability preserving reduced-dimension finite difference (RDFD) schemes for solving Caputo-type time-fractional diffusion equations (FDEs) of order α ∈ (0, 1) in one and two spatial dimensions. The Caputo derivative is discretized by the L2-1σ formula, and spatial derivatives are approximated using the central finite difference (FD), forming the L2-1σ-FD schemes as the foundational numerical frameworks. To reduce the computational complexity of high-dimensional FDEs, we further develop L2-1σ-RDFD schemes by integrating proper orthogonal decomposition (POD), achieving efficient dimension reduction while preserving numerical accuracy. We rigorously prove the unconditional stability and optimal convergence order of the proposed RDFD schemes. Numerical experiments confirm the theoretical results and demonstrate that the RDFD schemes significantly reduce computational cost without sacrificing accuracy compared to the original FD schemes. The proposed methods offer a general and efficient framework for numerically solving fractional diffusion models and are readily extensible to more complex fractional partial differential equations arising in interdisciplinary fields such as computational physics, engineering mechanics, and mathematical biology.
This paper is devoted to the analysis and numerical approximation of a class of nonlinear parabolic quasi-variational inequalities with solution-dependent nonlocal constraints. Unlike classical variational inequalities, the admissible set is determined by a nonlinear obstacle operator that depends explicitly on the evolving solution, resulting in a moving feasible region whose geometry changes throughout the evolution. This intrinsic coupling between the governing parabolic operator and the admissible set substantially increases both the analytical and computational complexity of the problem. We first investigate the fundamental analytical properties of the nonlocal obstacle operator and establish its monotonicity, continuity, Lipschitz stability, non-expansivity, and compactness under suitable structural assumptions. These results provide the analytical framework required for the study of the corresponding quasi-variational problem. By combining monotone operator techniques, compactness arguments, energy estimates, pseudomonotonicity methods, and nonlinear fixed-point theory, we establish the existence and uniqueness of weak solutions together with continuous dependence on the problem data. For the numerical approximation, we develop a fully discrete semi-implicit finite element method obtained by coupling a conforming spatial discretization with an implicit time-stepping procedure and an iterative obstacle projection algorithm that enforces the solution-dependent constraint at every temporal level. The fully discrete problem is shown to be well posed through discrete stability estimates, energy inequalities, and rigorous convergence analysis, leading to optimal a priori error estimates for the proposed approximation. Extensive two-dimensional numerical experiments are presented to validate the theoretical results. The computational study investigates spatial and temporal convergence, discrete energy evolution, nonlinear iteration behaviour, computational efficiency, algorithmic scalability, and robustness with respect to different nonlinear diffusion laws. The numerical results exhibit excellent agreement with the theoretical predictions, accurately preserve the solution-dependent obstacle constraint, and demonstrate that the proposed numerical framework provides an accurate, stable, and computationally efficient approach for solving nonlinear parabolic quasi-variational inequalities with nonlocal state-dependent constraints.
